Kinetic schemes

The treatment may be made more detailed by supposing that the rate-determining step is actually from species O in the OHP (at potential relative to the solution) to species R similarly located. The effect is to make fi dependent on the value of 2 and hence on any changes in the electrical double layer. This type of analysis has permitted some detailed interpretations to be made of kinetic schemes for electrode reactions and also connects that subject to the general one of this chapter. 

It has not been found possible to reconcile all these observations with a simple kinetic scheme. A major difficulty is that whilst the stoichiometric concentrations of nitric acid and of acetic acid can be varied independently, the actual concentrations of these species cannot, because of the existence of the equilibrium ... 

If the data follow the kinetic scheme presented here, the values of k, calculated for the different runs should be constant ... 

Dimethylformamide [68-12-2] (DME) and dimethyl sulfoxide [67-68-5] (DMSO) are the most commonly used commercial organic solvents, although polymerizations ia y-butyrolactoae, ethyleae carboaate, and dimethyl acetamide [127-19-5] (DMAC) are reported ia the hterature. Examples of suitable inorganic salts are aqueous solutioas of ziac chloride and aqueous sodium thiocyanate solutions. The homogeneous solution polymerization of acrylonitrile foUows the conventional kinetic scheme developed for vinyl monomers (12) (see Polymers). 

Although the decomposition rate of peroxide is thus increased, the consequent lowering of steady-state peroxide concentration leaves the effective rate unchanged in the simple peroxide cycle kinetic scheme (25). In real systems, at certain critical levels, a catalyst can become an inhibitor (2,180). 

A kinetic scheme and a potential energy curve picture ia the ground state and the first excited state have been developed to explain photochemical trans—cis isomerization (80). Further iavestigations have concluded that the activation energy of photoisomerization amounts to about 20 kj / mol (4.8 kcal/mol) or less, and the potential barrier of the reaction back to the most stable trans-isomer is about 50—60 kJ/mol (3). 

A general kinetic scheme for mechanism-based inactivators is outlined as follows ... 

The oxidation of Fe(CN)g to Fe(CN)g by peroxidisulfate, SjOg", can be monitored spectrophotometrically by observing die increase in absorbance at 420 nm, D 2o well-mixed batch system. Assume dial die kinetic scheme is ... 

A postulated reaction mechanism is a description of all contributing elementary reactions (we will call this the kinetic scheme), as well as a description of structures (electronic and chemical) and stereochemistry of the transition state for each elementary reaction. (Note that it is common to mean by the term transition state both the region at the maximum in the energy path and the actual chemical species that exists at this point in the reaction.)... 

Much experimental study suggests that this reaction is complex. The essential steps are shown in the following kinetic scheme. 

A third reaction, a fast proton transfer, then generates the equilibrium concentrations of RCOO and R OH. In this kinetic scheme I represents an intermediate, whose structure is shown in this amplified description ... 

The overall reaction stoichiometry having been established by conventional methods, the first task of chemical kinetics is essentially the qualitative one of establishing the kinetic scheme in other words, the overall reaction is to be decomposed into its elementary reactions. This is not a trivial problem, nor is there a general solution to it. Much of Chapter 3 deals with this issue. At this point it is sufficient to note that evidence of the presence of an intermediate is often critical to an efficient solution. Modem analytical techniques have greatly assisted in the detection of reactive intermediates. A nice example is provided by a study of the pyridine-catalyzed hydrolysis of acetic anhydride. Other kinetic evidence supported the existence of an intermediate, presumably the acetylpyridinium ion, in this reaction, but it had not been detected directly. Fersht and Jencks observed (on a time scale of tenths of a second) the rise and then fall in absorbance of a solution of acetic anhydride upon treatment with pyridine. This requires that the overall reaction be composed of at least two steps, and the accepted kinetic scheme is as follows. 

In the process of establishing the kinetic scheme, the rate studies determine the effects of several possible variables, which may include the temperature, pressure, reactant concentrations, ionic strength, solvent, and surface effects. This part of the kinetic investigation constitutes the phenomenological description of the system. 

In Chapter 1 we distinguished between elementary (one-step) and complex (multistep reactions). The set of elementary reactions constituting a proposed mechanism is called a kinetic scheme. Chapter 2 treated differential rate equations of the form V = IccaCb -., which we called simple rate equations. Chapter 3 deals with many examples of complicated rate equations, namely, those that are not simple. Note that this distinction is being made on the basis of the form of the differential rate equation. 

There is no general explicit mathematical treatment of complicated rate equations. In Section 3.1 we describe kinetic schemes that lead to closed-form integrated rate equations of practical utility. Section 3.2 treats many further approaches, both experimental and mathematical, to these complicated systems. The chapter concludes with comments on the development of a kinetic scheme for a complex reaction. 

Consecutive reactions involving one first-order reaction and one second-order reaction, or two second-order reactions, are very difficult problems. Chien has obtained closed-form integral solutions for many of the possible kinetic schemes, but the results are too complex for straightforward application of the equations. Chien recommends that the kineticist follow the concentration of the initial reactant A, and from this information rate constant k, can be estimated. Then families of curves plotted for the various kinetic schemes, making use of an abscissa scale that is a function of c kit, are compared with concentration-time data for an intermediate or product, seeking a match that will identify the kinetic scheme and possibly lead to additional rate constant estimates. 

Kinetic scheme XI has been discussed by many authors. 

This is the procedure From the postulated kinetic scheme we write the differential rate equations. Take the Laplace transforms of the differential equations. Solve the resulting set of algebraie equations for the transforms of the concentrations. Then take the inverse transforms to obtain the coneentrations as funetions of time. 

As an example, we take kinetic scheme IX, for which the differential rate equations are... 

Scheme XVI is more difficult to analyze than are any of the kinetic schemes treated earlier in this chapter, because it includes a second-order reaction.

Scheme XVlIl introduces another feature, for this is a cyclic kinetic scheme.

Equation (3-137) is a general result for this kinetic scheme it means that only five of the six rate constants are independent. 

When reactions other than first-order processes are included in the kinetic scheme, reactant concentrations may appear in the denominator of the rate equation. Scheme XIX is an example. 

The simplest kinetic scheme that can account for enzyme-catalyzed reactions is Scheme XX, where E represents the enzyme, S is the substrate, P is a product, and ES is an enzyme-substrate complex. 

The quantitative description of enzyme kinetics has been developed in great detail by applying the steady-state approximation to all intermediate forms of the enzyme. Some of the kinetic schemes are extremely complex, and even with the aid of the steady-state treatment the algebraic manipulations are formidable. Kineticists have, therefore, developed ingenious schemes for writing down the steady-state rate equations directly from the kinetic scheme without carrying out the intermediate algebra." -" ... 

We have seen that a kinetic scheme does not have to be very complex before explicit solutions for concentrations as functions of time become difficult or impossible to obtain. Even with those complex schemes for which solutions are possible, the... 

If an analytical solution is available, the method of nonlinear regression analysis can be applied this approach is described in Chapter 2 and is not treated further here. The remainder of the present section deals with the analysis of kinetic schemes for which explicit solutions are either unavailable or unhelpful. First, the technique of numerical integration is introduced. 

The procedure, in analyzing kinetic data by numerical integration, is to postulate a reasonable kinetic scheme, write the differential rate equations, assume estimates for the rate constants, and then to carry out the integration for comparison of the calculated concentration-time curves with the experimental results. The parameters (rate constants) are adjusted to achieve an acceptable fit to the data. Carpen-(ej-48. pp. 76-81 some numerical calculations. Farrow and Edelson and Porter... 

Kinetic schemes that include both first-order and second-order reactions possess an ambiguity related to the different dimensions of the rate constants. We wilt use Scheme XXI to examine this. 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

The method is quite effective, but is not widely used now because of the ubiquity of digital computers. Zuman and Patel - 36. show circuit designs for some kinetic schemes. Williams and Bruice made good use of the analog computer in their study of the reduction of pyruvate by 1,5-dihydroflavin. In this simulation eight rate constants were evaluated variations in these parameters of 5% yielded discemibly poorer curve fits. 

Sections 3.1 and 3.2 considered this problem Given a complex kinetic scheme, write the differential rate equations find the integrated rate equations or the concentration-time dependence of reactants, intermediates, and products and obtain estimates of the rate constants from experimental data. Little was said, however, about how the kinetic scheme is to be selected. This subject might be dismissed by stating that one makes use of experimental observations combined with chemical intuition to postulate a reasonable kinetic scheme but this is not veiy helpful, so some amplification is provided here. 

A postulated kinetic scheme is a model or hypothesis. In accord with Occam s razor, the model should be no more complex than is required to account for the... 

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

This manner of implicitly including the rate equations in the kinetic scheme is veiy convenient. It is amplified with the statement that when AX is acetyl chloride, ki/k i is very large and the reaction occurs essentially only via the I route. When... 

A kinetic scheme that is fully consistent with experimental observations may yet be ambiguous in the sense that it may not be unique. An example was discussed earlier (Section 3.1, Consecutive Reactions), when it was shown that ki and 2 in Scheme IX may be interchanged without altering some of the rate equations this is the slow-fast ambiguity. Additional examples of kinetically indistinguishable kinetic schemes have been discussed.The following subsection treats one aspect of this problem. 

The elements of water are omitted because the concentration of water cannot be varied and, therefore, is not explicitly included in the rate equation. A complete kinetic scheme requires only reactions 1, 2, 4, 7, and 9 the other reactions are superfluous. For example, the rate term k3[H2A][OH ] is equivalent to k4[HA-], as is the term kg[H ][A -]. 

In these circumstances a decision must be made which of two (or more) kinet-ically equivalent rate terms should be included in the rate equation and the kinetic scheme (It will seldom be justified to include both terms, certainly not on kinetic grounds.) A useful procedure is to evaluate the rate constant using both of the kinetically equivalent forms. Now if one of these constants (for a second-order reaction) is greater than about 10 ° M s-, the corresponding rate term can be rejected. This criterion is based on the theoretical estimate of a diffusion-controlled reaction rate (this is described in Chapter 4). It is not physically reasonable that a chemical rate constant can be larger than the diffusion rate limit. 

These ideas can exclude certain kinetic schemes as impossibilities. Thus, Scheme XVIII is possible, but Scheme XXIX is impossible, because B is formed directly from A, whereas A can only be regenerated from B via C, which is a different mechanism. 

---